Hi all, I have my exam in differential equations in one week so I will probably post alot of question. I hope you wont get tired of me!
Homework Statement
This is Legendres differential equation of order n. Determine an interval [0 t_0] such that the basic existence theorem guarantees a...
Homework Statement
Define the w-limit set (omega) of a point. Show that w(x) is closed.
Homework Equations
The Attempt at a Solution
The definition of a limit set is the set of points to which there exists a sequence t_n→∞ such that \phi(t_n,x) → y
The second question. I was...
\inHomework Statement
Denote by d(x,A) = inf |x-y|,y \in A, the distance between a point x \in R^n and a set A \subseteq R^n. Show
|d(x,A)-d(z,A)| \leq |x-z|
In particular, x → d(x,A) is continuous
Homework Equations
The Attempt at a Solution
I have no idea on how to prove this. I drew...
Homework Statement
Using that the Fourier transform of e^{|x|} is \frac{2}{\xi^2+1}. Compute the Fourier transform of \frac{x}{(x^2+1)^2}
Homework Equations
The Attempt at a Solution
My first thought was to try and rewrite the problem in a form I recognized, tried a couple of things but...
Homework Statement
Hi y'all, ran into some trouble with a fourier transform
Im supposed to find the fourier transform of
f(x)=\frac{1}{x^{2}+6x+13}
Homework Equations
Not that I know
The Attempt at a Solution
I tried integrating this with no luck.
All help is as usual...
Homework Statement
Determine the polynomial of the form P(x)=x^3+ax^2+bx+c
that minimizes
\int[P(x)]^2
Homework Equations
The Attempt at a Solution
My first thought was that I should find a second degree polynomial that minimizes x^3. That didn't work at all! So now I'm...
Homework Statement
u_{t}=3u_{xx} x=[0,pi]
u(0,t)=u(pi,t)=0
u(x,0)=sinx*cos4x
Homework Equations
The Attempt at a Solution
with separation of variables and boundry conditions I get:
u(x,t)= \sumB_{n}e^-3n^{2)}}*sinnx
u(x,0)=sinx*cos4x
f(x)=sinx*cos4x=\sumB_{n}*sinnx...
Homework Statement
Let f be the 2pi periodic function defined by f(x)=e^{cos(x^{2})}
for 0 < x < 2pi. What is the value of the Fourier series at x=4pi
Homework Equations
The Attempt at a Solution
I don't even know where to start.
All help is much appreciated!
Homework Statement
Solve inital value problem: x''+2x'+x=g(t)
g(t)=
t 0<t<1
2-t 1<t<2
0 t>2
Homework Equations
Second shift theorem, Heaviside function and Laplace transforms. I denote Heaviside,functuon H(t-a), and Laplace transform with L
The Attempt at a Solution
I...
Homework Statement
Evaluate \int\intT (x^2+y^2) dA, where T is the triangle with the vertices (0,0)(1,0)(1,1)
Homework Equations
The Attempt at a Solution
\int d\theta \int r^3 dr
Thats how far I got, not really sure about boundries on r. First integrals boundrie should be 0 to pi/4. Is...
This shouldn't be a problem but I just can't this right!
Homework Statement
Find the volume of the body that is generated when y= 1/x(sqrt(4-x^2)) ,1 <= x <2, is rotated around the y-axis
Homework Equations
Using cylindrical shells :
2 pi ∫x (g(x)-f(x)) dx
The Attempt at a Solution
2 pi...